1. 40
Resolving Multi Objective Stock Portfolio
Optimization Problem Using Genetic Algorithm
Hoklie
Department of Industrial Engineering
University Al Azhar Indonesia (UAI)
Jakarta, Indonesia
e-mail: indwr_87@yahoo.co.id
Lavi Rizki Zuhal
Department of Industrial Engineering
University Al Azhar Indonesia (UAI)
Jakarta, Indonesia
e-mail: lavirz@ae.itb.ac.id
Abstract— Portfolio optimization is an important research
field in modern finance. The most important characteristic
within this optimization problem is the risk and the returns.
Modern portfolio theory provides a well-developed paradigm
to form a portfolio with the highest expected return for a given
level of risk tolerance. Multi objective portfolio optimization
problem is the portfolio selection process that result highest
expected return and smallest identified risk among the various
financial assets.
In this paper, we propose to identify expected return (mean
profit) and risk using historical data of stock prices. The
downside values of the variance of each stock are considered to
be the identified risk in first case. The Value at Risk of each
stock that we obtained using parametric and historical
simulation methodology are considered to be the identified risk
in second case.
One of method being widely used lately for optimization
need is Genetic Algorithm or GA. This method adapted the
mechanism of biology mechanism involving natural selection.
With many advantages it has, numerical process of a portfolio
optimization in both case are attempted to be coupled with
Genetic Algorithm.
The values of expected return and risk of each stock will be
used as inputs into a fitness function in Genetic Algorithm.
Performance evaluation is done by examining the parameters
of GA, such as population size, stall generation and number of
elitism.
Keywords-multi objective portfolio optimization; genetic
algorithm; fitness function; expected return; risk
I. INTRODUCTION
Optimization of asset allocation is complex problem. It
is non-linear with many local optima [04]. Genetic
Algorithms (GA) is a heuristic random search that is based
on the evolutionary process of living beings. In a GA, we
encode the solution to the problem that we want to solve as
a “genome”, generate many of those solutions, test them
according to a given heuristic (the fitness function), and
generate new candidates from the best ranked “parents”. By
repeating this process, the algorithm is able to find good
solutions to a wide range of practical problem. The
difficulties during the optimization process that is involving
the complex formulae and sensitive boundary conditions
conceivably can be handled with this method [01].
This paper attempts to apply the advantages of Genetic
Algorithm to generate portfolio, with a certain criteria,
namely expected return and identified risk. The research
methodology of this paper is coding the numerical analysis
and optimization method by using MATLAB as language of
programming.
II. GENETIC ALGORITHM
In this paper we develop genetic algorithm code, listed
below are Genetic Algorithm components used in this
program:
· Encoding initialization: Real number encoding
· Cross over: One-point cut cross over
· Mutation: Gaussian mutation
· Evaluation: Linear fitness ranking
· Selection: Roulette wheel
· Termination criteria: Stall generation
· Elitism applied
The validity of developed program has been verified by
comparing to relevant examples. Several cases on
optimization that have been done to verify the performance
of developed Genetic Algorithm. Binary encoding gives
much more stable value of variables that are generated [10].
It is caused by the representation of gene code to the
solution have similar tendency meanwhile the real-number
encoding give the actual value. However, on the real-
number encoding, the result convergent is faster than binary
encoding.
In conclusion, real number encoding is much widely
used to solve the problem. Also, in this study, real number
encoding is used. Both verification results inform us that the
optimization program based on Genetic Algorithm has quite
well performance to be applied to the simple or complex
problem, with or without constraint. The next challenge is
how to apply developed genetic algorithm code to solve
portfolio optimization problem.
III. PORTFOLIO OPTIMIZATION
The multi objective portfolio optimization problems
have received increased interest from researchers with
various backgrounds since early 1952 [8]. A modern
portfolio theory provides a well-developed paradigm to
form a portfolio with the highest expected return for a given
level of risk tolerance. The mathematical model of the
978-1-4244-5585-0/10/$26.00 ©2010 IEEE

2. 41
Portfolio Optimization problem is due to Markowitz [8]. We
define a portfolio P as a set of N real valued weights
(w0,w1, ...wN) which correspond to the N available assets in
the market. The weights obey the following:
0
1
N
i
i
w
=
=å (1)
0 1iw£ £ (2)
Each asset has an expected return value, expressed by Ri.
The expected return value for the portfolio is given by the
sum of the expected return values for the assets:
1
n
W i i
i
r w r
=
= å (3)
Also, each asset has a risk measure, is . The risk of an
asset is defined as downside values of the variance of that
asset’s returns over time, and the risk of the Portfolio is
defined as:
1 1
n n
W i j ij
i j
w ws s
= =
= åå (4)
Where
2
ii is s= is the variance of ir , and ijs is the
covariance between ir and jr . The calculation for the error
of the portfolio in equation (4) can then be divided in two
parts, one containing the variance of the returns, and another
containing the covariance of the returns:
2 2
n
W i i i j ij
i i j
w w ws s s
¹
= +å å (5)
The first part of equation (5) describes the portfolio risk
component composed by the risk of individual assets. It is
called the specific risk. If there is no correlation between the
assets in the portfolio (the second part of the equation is 0),
we can show that the specific risk can be reduced by
diversification (adding more assets to the portfolio).
IV. PORTFOLIO OPTIMIZATION USING GENETIC
ALGORITHM
To start the analyses, we first have to conduct problem
definition. This section describes the development of
portfolio-optimization genetic algorithm program. Structure
of the program can be seen as a map by following flowchart
in figure 1. In the encoding procedure, the value of the gene
in the chromosome is generated randomly. When we are
generating the weight vector, we have to rescale the weight
to satisfy equation (1). As the result, we convert the weight
into
1
i
i N
i
i
v
w
v
=
=
å
.
In this section, we also show the experiments that were
performed to verify our proposition. We also carried out
simulation using real world historical data from Jakarta
Islamic Index in Indonesia market as a case study, and
observed good results with the proposed method.
Figure 1. Flowchart of Portfolio Optimization using Genetic Algorithm
Program.
A. Verification of Portfolio Optimization GA Program
1) Data Used
To perform the verification of GA code, we use the data
from reference [20] the paper entitled "Asset Allocation
using Goal Programming" written by Nichols & Ravindran.
In addition to using historical data from the paper, we also
use the same model to formulate a fitness function and using
the results for later comparison with the results of the GA
code.
2) Return and Risk Estimation
To determine the expected return and risk identified as
input for the fitness function, we use the Markowitz model
that also used in reference [20]. In the Markowitz model,
expected return is the average return from the historical
return data, while the identified risk is standard deviation
from the historical return data.
3) Fitness Function Development
In the optimization, objective function consideration is a
part that not less important than the developed program
itself. Evaluation function used for the GA is based on the
total expected return and the risk of the portfolio. For the
portfolio selection problem, we consider the total expected
return and the risk. The expected return is to be maximized,
while the risk is to be minimized. From the definitions and
results of the expected return and risk value from the
previous section, we can derive a utility function to use as
the Fitness function for the Genetic Algorithm. Fitness
function in this case is defined as:
0
0 1
.
( )
( )
. .
N
i i
i
N N
i j ij
i j
r w
f return
fitness
f risk
w w s
=
= =
= =
å
åå
(6)

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4) Results
Genetic Algorithm ends like other optimization
algorithms too, by testing for convergence. Figure 2
following shows fitness value plot, which resulting by
genetics algorithm on each generation. On that graph, blue
line represent fitness's value best individual, red line
represents fitness's average value individual on that
generation and black line represents fitness's value best
individual on processes random search (genetic algorithm
without evaluation). From that figure, we can see that
genetic algorithm give the results that convergent to the
better solution, in consequence we can said that genetic
algorithm is successful applied in this case.
0 20 40 60 80 100 120 140
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Performance of GA
best value GA:blue, best value RS:black, mean value GA:red
number of generations
value
Figure 2. Convergence Check for Nichols & Ravindran Case with
PopSize = 100, ST = 20 and Number Elitism = 10.
After convergence check done, hereafter programs to be
run with various combination parameters. Parameters that
are combined in this paper are Population Size, Stall
Generation and Elitism. Results from a combination of
parameters can be seen in table 1. Table 1 show that the
time needed to run the program ranges between 10 - 40
seconds, so that we can say this program is fast enough to
produce a solution. The time required to run the program in
general with the proportionate number of evaluation
function. Best value for each performance criteria, the cells
marked by blue.
However, the criteria for portfolio performance namely,
portfolio return, portfolio risk and the resulting fitness value
does not have a clear relationship of each of the criteria or a
combination of its parameters. Solution that provides the
best results on the performance of a portfolio will be
compared with the solutions generated by other methods.
Comparison of the resulting solution with the GA and other
methods, can be seen in figure 3.
TABLE I. GA PERFORMANCE FOR NICHOLS & RAVINDRAN CASE
UNDER VARIED PARAMETER
Parameter Performance
Num
Pop
ST Elitism
Portfolio
Return
Portfolio
Risk
Fitness
Value
Elapsed
Time
(Seconds)
Number of
Evaluation
Function
10 11,251 14,515 0,77511 19,327 14700
20 11,260 14,273 0,78886 20,595 1560020
30 12,304 16,053 0,76647 29,570 20900
10 11,118 15,121 0,73530 7,489 16500
20 11,258 15,668 0,71857 37,404 19700
100
30
30 11,366 15,559 0,73048 41,848 21800
10 12,259 17,848 0,68686 33,076 28200
20 11,634 14,521 0,80115 26,784 3020020
30 11,247 15,126 0,74358 35,292 29800
10 10,718 15,858 0,67587 8,053 33200
20 11,215 15,326 0,73178 8,218 38000
200
30
30 11,487 15,537 0,73929 27,532 36600
Figure 3. Comparison Result FGP, CGP, and GA on Nichols &
Ravindran Case.
B. Case Study: Jakarta Islamic Index
In this section, we will try to apply the GA code to
complete the portfolio optimization problem using the real
data and the model is quite adequate. In this case study we
use data from stocks listed on the Jakarta Islamic Index in
April 2009.
1) Data Used
The suitable way of constructing a portfolio is to select
some good quality assets first and then to optimize asset
allocation using portfolio optimization model. To select
some good quality asset/stocks, we use the Syariah criteria
stocks. So, we can use stocks data listed on the Jakarta
Islamic Index (JII).
In this paper, we propose to identify expected return
(mean profit) and risk using historical data of stock prices.
The daily data used in this paper is stock closing price for
the last five years (April 2004-2009), obtained from Jakarta
Stock Exchange for Jakarta Islamic Index Stock.

4. 43
2) Return and Risk Estimation
To identify expected return we use mean profit for each
stocks during last five years. The downside values of the
variance of each stock which we obtained using parametric
and historical simulation methodology (average) are
considered to be the identified risk.
Risk calculator program developed by Anatoly Ivanov
[32] used to calculate the portfolio return and identified risk
for the stocks listed on the Jakarta Islamic Index in April
2009.
3) Fitness Function Development
In this case, we will use adaptive weight sum approach
[07] to formulate fitness function for Multi Objective
Genetic Algorithm. For a given generation, two extreme
points are identified and the adaptive weights are calculated
as:
1 max min
1 1
1
p
z z
=
-
(7)
2 max min
2 2
1
p
z z
=
-
(8)
Fitness function for Jakarta Islamic Index case is defined
as:
1
01 1
2 2
2
0 1
. .
.( )
( ) .
. . .
N
i i
i
N N
i j ij
i j
p r w
p zf return
fitness
f risk p z
p w w s
=
= =
= = =
å
åå
4) Results
As on previous case, after convergence check done,
hereafter programs to be run with various combination
parameters. Results from a combination of parameters can
be seen in table 2.
TABLE II. GA PERFORMANCE FOR JAKARTA ISLAMIC INDEX CASE
UNDER VARIED PARAMETER
Parameter Performance
Num
Pop
ST Elitism
Portfolio
Return
Portfolio
Risk
Fitness
Value
Elapsed
Time
(Seconds)
Number of
Evaluation
Function
10 14,1987 357,678 2,65868 20,552 14900
20 14,950 378,907 2,64253 29,898 2030020
30 15,752 377,006 2,79829 29,419 16600
10 15,308 380,496 2,69451 45,367 32000
20 14,937 367,377 2,72301 36,342 24500
100
30
30 14,398 370,544 2,60234 26,113 25900
10 14,111 353,080 2,67671 36,084 52800
20 16,832 406,716 2,77176 33,872 5660020
30 15,783 379,677 2,78413 29,198 51400
10 13,828 345,274 2,68223 49,538 56600
20 15,016 363,988 2,76298 36,616 60000
200
30
30 16,445 384,315 2,86588 63,492 55200
5) Parameter Observation
Figure 4 shows the plot between the performances of
genetic algorithm with each of the parameters observed.
From that figure, does not appear that there is a clear
relationship between the parameters and the resulting fitness
value, so we can say that there is no 'trend' that occurs.
From the observation of the parameters in both these
cases, it can be observed that the parameter does not have a
clear relationship with the resulting performance. Thus, the
search parameters should be done to give the most optimal
solution with run the program under various combinations
of parameters.

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Figure 4. Top: Elitism, Middle: Stall Generation, Bottom: Number
Population Parameter Observation on Jakarta Islamic Index Case.
V. CONCLUSIONS AND FUTURE WORKS
Some of the conclusions based on the three previous
chapters are as follow:
· Genetic Algorithm code that developed in this paper
can provide good results for simple optimization
problem and portfolio optimization problem.
· Formulate portfolio optimization problem can be
done using Markowitz Model, Risk & Return
Estimation Method and Multi Objective Genetic
Algorithm (MOGA) as in the formulation results in
this paper.
· For solve portfolio optimization using GA, we
required weight generating value procedure.
· Fitness function that developed in both case are able
to be used as an input Genetic Algorithm.
· To find the optimal solution and optimize GA
performance, we must observe sufficient GA
parameters.
The whole process in finishing this paper gives much
information that can be used for the next investigation
process and GA application. Some ideas that can be
continued for the next investigation are just like following
points,
· Future research will investigate new strategies for
improving the performance.
· Research can be focused on developing a hybrid
form of algorithms, which combine the good
features of different algorithms, says Genetic
Algorithms and Tabu Search.
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